Charles Loewner

Charles Loewner
Born	(1893-05-29)29 May 1893 University of Prague
Doctoral advisor	Georg Alexander Pick
Doctoral students	Lipman Bers William J. Firey Adriano Garsia Roger Horn Pao Ming Pu

Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.

Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner.^[1][2]

Loewner received his Ph.D. from the

. One of his central mathematical contributions is the proof of the

P. M. Pu

.

Loewner's torus inequality

In 1949 Loewner proved his torus inequality, to the effect that every metric on the 2-torus satisfies the optimal inequality

${\displaystyle \operatorname {sys} ^{2}\leq {\frac {2}{\sqrt {3}}}\operatorname {area} (\mathbb {T} ^{2}),}$

where sys is its systole. The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called equilateral torus, i.e. torus whose group of deck transformations is precisely the hexagonal lattice spanned by the cube roots of unity in ${\displaystyle \mathbb {C} }$.

Loewner matrix theorem

The Loewner matrix (in

(of real ${\displaystyle C^{1}}$ functions) associated with 2 input parameters consisting of (1) a real function on a subinterval of the real numbers and (2) an ${\displaystyle n}$-dimensional vector with elements chosen from the subinterval; the 2 input parameters are assigned an output parameter consisting of an ${\displaystyle n\times n}$ matrix.^[3]

Let ${\displaystyle f}$ be a real-valued function that is continuously differentiable on the

${\displaystyle (a,b)}$.

For any ${\displaystyle s,t\in (a,b)}$ define the divided difference of ${\displaystyle f}$ at ${\displaystyle s,t}$ as

${\displaystyle f^{[1]}(s,t)={\begin{cases}\displaystyle {\frac {f(s)-f(t)}{s-t}},&{\text{if }}s\neq t\\f'(s),&{\text{if }}s=t\end{cases}}}$.

Given ${\displaystyle t_{1},\ldots ,t_{n}\in (a,b)}$, the Loewner matrix ${\displaystyle L_{f}(t_{1},\ldots ,t_{n})}$ associated with ${\displaystyle f}$ for ${\displaystyle (t_{1},\ldots ,t_{n})}$ is defined as the ${\displaystyle n\times n}$ matrix whose ${\displaystyle (i,j)}$-entry is ${\displaystyle f^{[1]}(t_{i},t_{j})}$.

In his fundamental 1934 paper, Loewner proved that for each positive integer ${\displaystyle n}$, ${\displaystyle f}$ is ${\displaystyle n}$-monotone on ${\displaystyle (a,b)}$ if and only if ${\displaystyle L_{f}(t_{1},\ldots ,t_{n})}$ is

for any choice of ${\displaystyle t_{1},\ldots ,t_{n}\in (a,b)}$.^[3][4][5] Most significantly, using this equivalence, he proved that ${\displaystyle f}$ is ${\displaystyle n}$-monotone on ${\displaystyle (a,b)}$ for all ${\displaystyle n}$ if and only if ${\displaystyle f}$ is real analytic with an analytic continuation to the upper half plane that has a positive imaginary part on the upper plane. See Operator monotone function.

Continuous groups

"During [Loewner's] 1955 visit to Berkeley he gave a course on

In Loewner's terminology, if ${\displaystyle x\in S}$ and a

group action

is performed on ${\displaystyle S}$, then ${\displaystyle x}$ is called a quantity (page 10). The distinction is made between an abstract group ${\displaystyle {\mathfrak {g}},}$ and a realization of ${\displaystyle {\mathfrak {g}},}$ in terms of denoted ${\displaystyle J({\overset {u}{v}})}$ (page 41). The term invariant density is used for the have equal left and right invariant densities (page 48).

A reviewer said, "The reader is helped by illuminating examples and comments on relations with analysis and geometry."[9]

See also

• Löwner-John ellipsoid
• Schramm–Loewner evolution
• Loop-erased random walk
• Systoles of surfaces

References

• Berger, Marcel: À l'ombre de Loewner. (French) Ann. Sci. École Norm. Sup. (4) 5 (1972), 241–260.
• Loewner, Charles; Nirenberg, Louis: Partial differential equations invariant under conformal or projective transformations. Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York, 1974.

External links

• Stanford memorial resolution
• O'Connor, John J.; Robertson, Edmund F., "Charles Loewner", MacTutor History of Mathematics Archive, University of St Andrews